3.1 \(\int \sqrt {1+2 x} \, dx\)

Optimal. Leaf size=13 \[ \frac {1}{3} (2 x+1)^{3/2} \]

[Out]

1/3*(1+2*x)^(3/2)

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {32} \[ \frac {1}{3} (2 x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + 2*x],x]

[Out]

(1 + 2*x)^(3/2)/3

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {align*} \int \sqrt {1+2 x} \, dx &=\frac {1}{3} (1+2 x)^{3/2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ \frac {1}{3} (2 x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + 2*x],x]

[Out]

(1 + 2*x)^(3/2)/3

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fricas [A]  time = 0.39, size = 9, normalized size = 0.69 \[ \frac {1}{3} \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^(1/2),x, algorithm="fricas")

[Out]

1/3*(2*x + 1)^(3/2)

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giac [A]  time = 0.01, size = 9, normalized size = 0.69 \[ \frac {1}{3} \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^(1/2),x, algorithm="giac")

[Out]

1/3*(2*x + 1)^(3/2)

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maple [A]  time = 0.03, size = 10, normalized size = 0.77 \[ \frac {\left (2 x +1\right )^{\frac {3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+2*x)^(1/2),x)

[Out]

1/3*(1+2*x)^(3/2)

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maxima [A]  time = 0.56, size = 9, normalized size = 0.69 \[ \frac {1}{3} \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^(1/2),x, algorithm="maxima")

[Out]

1/3*(2*x + 1)^(3/2)

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mupad [B]  time = 0.32, size = 9, normalized size = 0.69 \[ \frac {{\left (2\,x+1\right )}^{3/2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 1)^(1/2),x)

[Out]

(2*x + 1)^(3/2)/3

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sympy [A]  time = 0.10, size = 8, normalized size = 0.62 \[ \frac {\left (2 x + 1\right )^{\frac {3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)**(1/2),x)

[Out]

(2*x + 1)**(3/2)/3

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